(a)        (i)         Construct an internal tangent to two unequal circles of diameter 30
                                    and 58 respectively, whose centres are 100 apart.
                       
(ii)        Measure and state the distance between the points of tangency.

 

            (b)

   

1. Fig. 3 shows a pentagon PQRST: PQ = 30, QR = 35,

RS = 56, ST = 40 and TP = 20.  Enlarge the pentagon to a ratio of 3:1.
           

1. Measure and state the enlarged length RS.

 

Candidates were expected to:

(a)        INTERNAL TANGENT TO TWO UNEQUAL CIRCLES:
            (i)         draw two vertical centre lines at A and 100 apart;
            (ii)        draw one horizontal centre line;

1. draw two unequal circles Ø30 and  Ø58 at A and B respectively;
2. draw another circle Ø88 (Ø58 + Ø30) at centre B;
3.  bisect AB and draw semi-circle on AB;
4.  join BD to intersect circle Ø58 to locate point T;
5.  draw a line at A parallel to BD to the smaller circle Ø30 to locate point P;
6.  join PT to obtain the internal tangent;
7.  measure and state the length of tangent PT = 91 ± 1.

 

The solution is shown below:

(a)        ALTERNATIVE METHOD
                        Candidates were expected to:

1. draw two vertical centre lines at A and B 100 apart;
2. draw one horizontal centre line;
3. draw two unequal circles Ø30 and Ø58 at A and B respectively;
4. draw another circle Ø88  (Ø58 + Ø30) at centre B;
5. bisect AB and draw semi-circle on AB to intercept circle Ø88 at D;
6. join BD to locate point T on circle Ø58;
7. draw a line parallel to BD from A to locate point P on circle Ø30;
8. join PT to obtain the internal tangent;
9. measure and state points of tangency PT.

The solution is shown below:

(b)        CONSTRUCTING PENTAGON PQRST (BY AREA)
            Candidates were expected to:

1. construct the pentagon PORST;
2. extend PQ to the right;
3. divide PQ extended into three (3) equal parts;
4. construct semi-circles at P1 and P3;
5. drop a perpendicular at, Q intercept the semi-circles at points 1 and 3;
6. extend QP to the left;
7. join point 1 to P and draw parallel line from point 3 to intercept line QP extended at P1;

1. radiate lines from Q through T, S and R;
2. draw construction lines P1 T1, T1 S1  and S1  R1 parallel to PT, TS and SR respectively
3. complete the enlarged pentagon P1  QR1  S1  T1 ;

 

1. measure and state the length of R1  S1  

The solution is shown below:

 

 

3.         (b)        ALTERNATIVE METHOD 1 (BY SIZE)
                        CONSTRUCTING THE PENTAGON PQRST
                        Candidates were expected to:

1. construct the pentagon PQRST;
2. extend PQ to the right;
3. draw a line at P at an angle to PQ and divide the line into three equal parts;

1. radiate lines at P through T, S and R;

 

1. join point 1 to Q and draw parallel line from point 3 to intercept PQ extended at Q1;

1. draw construction lines Q1 R1, R1 S1 and S1 T1 parallel to QR, RS and ST respectively;

 

1. complete the enlarged pentagon PQ1  R1  S 1  R1  T1 ;

1. measure and state the length S 1  R1 .

The solution is shown below:

CONSTRUCTING THE PENTAGON PQRST

Candidates were expected to:

1. construct the pentagon PORST;
2. extend PQ to the right;
3. draw a line at P at an angle to PQ and divide the line into three equal parts;
4. radiate lines at P through T, S and R;
5. join point 1 to Q and draw parallel line from point 3 to intercept PQ extended at Q1;

 

1. draw construction lines Q1 R1 , R1S1  and S 1T1 parallel to QR, RS and ST respectively;

1. complete the enlarged pentagon PQ1  R1  S1  R1  T1 ;

 

1. measure and state the length S1 R 1.

 

 

 

 

 

 

 

 

 

 

The solution is show below

3(b)      ALTERNATIVE METHOD2 (BY SIZE)